Here's a list of museums, libraries, and cultural sites that are (at least partially) related to the Native American flute. The list organized by location. Most of these links opens in a separate browser window. This list incorporates the institutions referenced in [Higgins 2007].

Arkansas

California

Kenneth G. Fiske Museum of Musical Instruments — Claremont.

The Phoebe Apperson Hearst Museum of Anthropology (formerly the Robert H. Lowie Museum of Anthropology) at the University of California - Berkeley.
One of the major collections of ethnographic sound recordings
in the United States. According to [McAllester 1996], this museum holds 2,712 items on wax cylinders and many additional recordings on discs, wire, and tape.

Connecticut

District of Columbia (Washington, D.C.)

The National Museum of the American Indian / Smithsonian Institution.
Established by an act of Congress in 1989, the museum works in collaboration with the Native peoples of the Western Hemisphere to protect and foster their cultures by reaffirming traditions and beliefs, encouraging contemporary artistic expression, and empowering the Indian voice.

Illinois

The Field Museum — Chicago.
Visited on August 4, 2016 and photographed the flutes and artifacts that were on display. The museum did not grant me access to any resources, materials, or publications in their collection.

The National Anthropological Archives, Smithsonian Institution — Suitland.
The National Anthropological Archives and Human Studies Film Archives collect and preserve historical and contemporary anthropological materials that document the world's cultures and the history of anthropology. Many photographs of indigenous cultures, and some relevant flute photos.

New Jersey

New Mexico

Museum of Indian Arts and Culture — Santa Fe.
One of four museums in the Museum of New Mexico system, is a premier repository of Native art and material culture and tells the stories of the people of the Southwest.
Houses an example of the Taos style of flute.

North Carolina

The Museum of the Cherokee Indian — Cherokee.
Located on tribal land in the Great Smoky Mountains, this museum maintains a collection and photo archive database. Their on-line digital archive shows an interesting double flute and several relevant images.

Nebraska

Nevada

Lost City Museum — Overton.
Built by the National Park Service to exhibit artifacts that were being excavated from Pueblo Grande de Nevada.

Oklahoma

The Museum of the Red River — Idabel.
Housing over 20,000 objects in its collections, the Museum sponsors exhibits, lectures, and other programs, and supports ongoing research efforts in the study of American native peoples.

Oregon

Pennsylvania

The American Philosophical Society Library — Philadelphia.
A large collection of Native American images, recordings, and correspondence of prominant anthropoligists. Some of the collection is digitized and available on-line.

Rhode Island

Haffenreffer Museum of Anthropology, Brown University — Providence.
Collection contains a Kiowa instrument dated between 1875–1910, described as having seven finger holes. Photo and description of flute in Hau Kola!
The Plains Indian Collection of the Haffenreffer Museum of Anthropology, Brown University by Barbara Hail ([Hail 1980], page 184, length 57cm, Collected by O. Fulda (information from [Higgins 2007]).

Collections Research Center, Haffenreffer Museum of Anthropology, Brown University — Bristol.
The Haffenreffer Museum of Anthropology's Collections Research Center holds more than 100,000 ethnographic objects, archaeological specimens and images from all parts of the world, with particular strengths in the Americas, Africa and Southeast Asia. These collections include nearly 60,000 objects collected by Rudolf F. Haffenreffer before his death in 1954 and a similar number obtained through fieldwork, donations, and strategic acquisitions since the Museum became part of Brown University in 1955.

South America

Argentina

Museo Etnográfico Juan B. Ambrosetti — Buenos Aires.
From Fodor's Argentina, 2012:
The ground-floor galleries trace the history of human activity in Patagonia, with an emphasis on the tragic results of the European arrival. Dugout canoes, exquisite Mapuche silver jewelry, and scores of archive photos and illustrations are the main exhibits.
In the upstairs northwestern Argentina gallery the focus is mainly archaeological. Displays briefly chronicle the evolution of Andean civilization, the heyday of the Inca empire, and postcolonial life. Artifacts include ceramics, textiles, jewelry, farming tools, and even food: anyone for some 4,000-year-old corn?
The museum was closed when I visited on January 24, 2013.

Museo de la Pategonia — Bariloche de los Andes.
From Fodor's Argentina, 2012:
This museum tells the social and geological history of northern Patagonia with displays of Indian and gaucho artifacts and exhibits on regional flora and fauna. The histories of the Mapuche and the Conquista del Desierto (Conquest of the Desert) are explained in detail.
The museum has a limited display on the first peoples of the region.
There were no music artifacts on display when I visited on January 18, 2013.

Museo Pobladores (Pioneer Museum) — San Martin de los Andes.
From Fodor's Argentina, 2012:
The museum is a tiny building next to the tourist office that was the original city council lodge. It is mainly dedicated to Mapuche ceramics and weavings, and a collection of 13,000-year-old tools and fossils gives an idea of ancient life in the region.
The museum was closed when I visited on January 21, 2013.

Chile

Museo Chileno de Arte Precolombino — Santiago.
More than 1,500 pre-Columbian artifacts on displya, including a flute of the Nasca culture. Limited visit on January 11, 2013 during their reconstruction and expansion.

Belgium

Musée des Instruments de Musique (Musical Instrument Museum) — Brussels.
A huge collection of musical instruments, including rare and unusual instruments such as the a case of Slovakian Fujaras. However, no Native American flutes are on display, nor were any in their collection when I asked during my visit on February 24, 2010.

Africa

Tunisia

Dar Ennejma Ezzahra Centre of Arab & Mediterranean Music — Sidi Bou Said.
A large museum with a permenant exhibition of musical instruments, a musical instrument making workshop, and an extensive sound library. Web site in English, Français, and Arabic. No Native American flutes are on display, nor were any in their collection when I asked during my visit on March 2, 2011, but a visit to this museum is still well worth your time if you are in the area.

Oceana

New Zealand

Auckland War Memorial Museum / Tamaki Paenga Hira — Auckland.
A large museum with an extensive collection of Māori artifacts.
No Native American flutes on display or in their collection when we visited on January 2, 2014, but they have an extensive collection of Māori musical instruments,
including examples of nguru, kōauau, pōrutu, and pūtōrino flutes.